An exact power series representation of the Baker–Campbell–Hausdorff formula

An exact representation of the Baker–Campbell–Hausdorff formula as a power series in just one of the two variables is constructed. Closed form coefficients of this series are found in terms of hyperbolic functions, which contain all of the dependence on the second variable. It is argued that this exact series may then be truncated and be expected to give a good approximation to the full expansion if only the perturbative variable is small. This improves upon existing formulae, which require both to be small. Several different representations are provided and emphasis is given to the situation where one of the matrices is diagonal, where a particularly easy to use formula is obtained.